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Example 1. (#12, lesson 4.2)
| Sketch the graph of a function on [-1,2] that has an absolute maximum but no local maximum. | Given Problem. #12a, lesson 4.2 |
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Red lines simply help us to see the interval which the question asked us about [-1,2]. For there to be no local maximum, we can't have any place on the graph where there would be a horizontal tangent line to the curve. One easy answer to this is simply a straight line that is neither horizontal nor vertical. We still have an a an absolute maximum. On the given interval, this would occur at x = 2, and the absolute maximum would be f(x) = 4. |
| Sketch the graph of a function on [-1,2] that has a local maximum but no absolute maximum. | Given Problem. #12b, lesson 4.2 |
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The Red lines again help us to see the interval which the question asked us about [-1,2]. The right red line is also an asymptote. We have a local maximum at approximately x = 1, and because of the asymptote we have no absolute maximum. |
Example 2. (#14, Lesson 4.2)
| Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. | Given Problem: #14a, lesson 4.2 |
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You can see that there are two locations where the slope of the tangent is zero, and there are local maxima. Likewise, you can see the point where there is one local minimum. There can be no absolute minimum because of the asymptotes at x = -5 and at x = -1 |
| Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers. | Given Problem: #14b, lesson 4.2 |
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There can be no local maxima at x = 3 because of the open dot at (3,2). Remember a critical number occurs when the derivative is zero or does not exist. In this sketch there are 5 places where the derivative is zero, and there are an additional 2 places where the derivative does not exist. (x = -6 and x = 3) |
Example 3. (#22, Lesson 4.2)
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Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f.
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Given Problem. (#22, Lesson 4.2) |
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The top portion of the sketch comes from the transformation of the square function. We are to 1)flip the graph up-side-down, and 2)move it up 2 units. Absolute Maximum: 2 .................Absolute Minimum: none Local Maximum: 2.......................Local Minimum: none |
Example 4. (#30, Lesson 4.2)
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Find the critical numbers of the function:
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Given Problem. (#30, Lesson 4.2) |
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Another form of the function which is easier to differentiate. |
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Now we differentiate, making sure to use the chain rule. |
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To find critical numbers we first need to find out when G'(x) can equal zero. This can only happen when the numerator is equal to zero. So one of our critical numbers is one half. |
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The other thing we need to look at for critical numbers is when the derivative does not exist. If the denominator of G'(x) was zero, this too would give us critical numbers. There are two solutions for x that would cause the denominator to be 0. |
| Critical numbers: x = 1/2, 1, and 0 | These then are the critical numbers of the function. |
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1-53, Odds
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